Multiplication and division linkage operator



Sept. 22, 1953 E. w. PIKE 2,652,978

MULTIPLICATION AND DIVISION LINKAGE OPERATOR Filed March 14, 1951 I 2 Sheets-Sheet l.

' INVEN T01? EUGENE W PIKE Sept. 22, 1953 E. w. PIKE 1 2,652,978

MULTIPLICATION AND DIVISION LINKAGE OPERATOR Filed March 14, 1951 2 Sheets-Sheet 2 lNVENTO/Z EUGENE IN. PIKE azwt ATOREV Patented Sept. 22, 1953 MULTIPLICATION AND DIVISION LINKAGE OPERATOR Eugene W. Pike, Newton, Mass., assignor to Raytheon Manufacturing Company, Newton, Mass., a corporation of Delaware Application March 14, 1951, Serial No. 215,515

2 Claims. 1

This invention relates to improvements in multiplication and division linkage operators and more particularly to those of the type disclosed in thecopending application, Serial vNo. 190,456, filed October 18, 1950, by Sumner D. Lewis.

In linkage operators of this type one factor of the multiplication is entered by a linear displacement of the first input element which is communicated to a second input element by a link. The angular displacement of the second input element represents the other factor of the multiplication. The total resultant linear displacement of a slide member mounted in a slot on the second input member represents the product. This displacement is transmitted to a slide mounted in a slot on a fixed frame member by a pivotally mounted link connecting the two slide members. The displacement may be more accurately transmitted to an output slide by a second pivoted link connected to the center point 01' the first output link and half its length, and. joined at its other end to a slide in the output slot as described more fully in the copending application of Philip T. Nickson, Serial No. 212,912,

filed February 27, 1951.

In both these cited applications the center line of the input slot is shown at right angles to that of the output slot. This construction permits minimum error when the range of at least one of the factors varies approximately symmetrically about the zero reference line. When the range of both factors represents values predominantly positive or negative, the errors are no longer at a minimum. It has been found that the errors may again be reduced to a minimum if the center line of the fixed input slot is positioned at the center of the anticipated swing of the angular displacement input member.

This modification in design leaves a small residual error that may be reduced further by adding a two-bar linkage to the output. Such a two-bar linkage comprises an additional slide mounted in a slot whose center line is positioned at an angle to the center line of the original output slot. This additional Slide is pivotally connected by a link to the original output slide. The relative angle of the output slides are chosen, together with the length of the link, to introduce a slight nonlinearity into the output to compensate for the remaining error of the improved linkage operator.

As before, this improved machine may be used for division by inserting the dividend as a displacement of the original output slide and inserting the divisor by displacing either the original input slide or the rotating input member with the resulting displacement of the unused member reprsenting the output.

These, and other advantages, features and objects of the invention will become more apparent from the following description taken in connection with the illustrative embodiments in the accompanying drawings, wherein:

Fig. 1 is an illustrative schematic diagram of one embodiment of the invention;

Fig. 2 is a diagrammatic illustration of the principle of the invention; and

Fig. 3 is a diagrammatic illustration of a modification of the invention utilizing a two-bar correction linkage added to the output.

In Fig. 1 the various members are shown with a particular set of dimensions. However, it should be understood that such a linkage arrangement will act as an approximate multiplier or divider for a wide selection of linkage dimensions. Therefore, the dimensions used are intended to be illustrative only, and not restrictive. The illustrative embodiment in Fig. 1, when used as a multiplier, has a first input member I!) and a second input member 12. The first input member I0 is mounted to slide within a slot l3 in a frame member M. The second input member I2 is pivoted to the frame I5 at point [6 and is formed with a slot l1 within which a slide [8 is mounted. A link 20 is pivotally fastened to slides I0 and 18 at points 2| and 22, respectively. The

linear displacement of point 22 is communicated to a slide 23 fitted in a slot 24 in a frame member 25 by a link 26 pivotally attached to a slide H3 at a point 22 and to the slide 23 by pin 21.

At the center of link 26 a third link 28 is pivotally attached by pin 30. This link 28 is attached at its other end to a slide 3| in the slot 24 in frame member 25 by a pin 32.

The dimensions are selected for the various parts by the design procedure described in the cited copending Lewis application. It is not thought necessary to repeat a description of this design procedure at this point.

As more fully described in the copending Lewis application referred to above, one factor of a multiplication is represented by the linear displacement of slide 10 from the position represented by line 33 to that represented by line 33', a distance represented by the dimension 34. The second factor is represented by the angular displacement 0 of the pivoted input l2. The result is to displace the slide 18 both angularly and linearly along the slot I1. This displacement is communicated to the slide 23 by the link 26 and 3 moves the pin 21 from the position represented by the line 35 to that represented by the line 36, a distance represented by the dimension 31.

If the center line of the slot [3 is positioned along a line passing through the pivot I6 of the rotatable input member l2, but not at right angles to the center ,line "of the output slot 24, ;certain errors will be introduced :by the angle ,the ,center line of the slot [3 makes with the center line of the slot [1 in the input member 12. This will be more clearly understood from a consideration of Fig. 2 which is a diagrammatic representation 'of the geometry of the part of the system including the input member I2, the input slide M1 and 21125 slot 13, and the link 20. The :pivot'point tithe center of the pin 2i on the slide [0, and the center of the pin 22 on the slide I 8 form the apexesnf a triangle. A line 40 is erected at point l6 perpendicular to the center line of the output slide 224. The angle between the perpendicular 40 and the center line of the slot I:3 isihereinafter-designated as :0. Thelength of the link :20 between points 2| an 22 is hereinafter designated as'a. Thezdistance of thepoint .22 .fromthe pivot l-6.'is hereinafterdesignated'as b. .The projection of this line H of length b on the slot \24 is y,-as shown bysthe perpendicular dropped from th'e point=22 to the center line of the output :slot '24. angular displacementlo'f the p'iovta-l input member I1 from the reference line 401-is represented :by The angle between'the centerrline 'of the slot I1 and the center line of the inputcslot 13 represented by A. The distance from the pivotpoint Hi to the slide 21 along the slot 13 is represented by a-i-zc, the length :of the link 20 ;plus :the displacement of the-slide 2| along theislot i3. Theangle between the link 20 :and the :center line :of :the input slot 13 is represented :by 13. The supplement .of the angle :between theacenter line mf the slot 11 and the link 20 fiszrepresented by C. *Eihe angular displacement :or the pivoted iinput member l2 from "the ireference line :is :represented by :The projection of thelmdinensionzhon-themeference line A!) is represented .by .:d. The output .1! can the written in two ways,

and 1% converting cos As to sinitfs in (4') Land :simplifying cos A= sin a "(5') It willbe recognized'that the expression /lsin .A

can be expanded to the series also can be recognized as expandable to the series:

'4 2)SlllA is substituted for the a: of the standard form.

Substituting the appropriate series in Equation 5 which can be furthersimplified to b 4421 +%(1 sm A+%'(1 1] substituting "(8) in (1') y=x sin @[1 +E sin hg- 111113113118 :terms between the square brackets :of Equation 9 define the terms withinttheibrackets in Equation 41 .and permit the error of a=given multiplier to be calculated to :anyudegree of accuracy required, or conversely the .link'ilength 'a may be :adjusted tso that, over the desired range of inputs-:1: rand qbythe error will not exceed the maximumspermissible-error for the particular-application. :It will also :be ,noted "that, with the dimensions computed according 'to this formula, will not be :afactor and'so this :formula is'used in designing the computer with the slot input slide.

' :Itwill be seen "from Fig. 2 and trigonometric principles that, if :a perpendicular 42 is drawn frompointfl to line 40 and another perpendicular line 43 is drawn from the point 22 to line '40, the distance d will be the difference between the projection of the distance between points 16 and 2| :along the slide l3 onto line =40 and the projection 10f the length of the link 20 on line 40.

The projection of a+m on the'line 40-is equal'to accae'ra' This reduces to: 5

d=(a+a:)cos 9-0. cos(-B) (l1)- =(a+x)cos 0-a(cos 0 cos B+sin 9 sin B) (12) The three angles of the triangle are A, B, and 180C A+B+180C'= 180 so that substituting (14) in (16) and converting cosines to sines:

cos B= It can be shown that cos B= g sin A+(1- sin A) 2 [1--Z 1- ?Sin A) S111 211% (18) so This can be expanded to the series cos B=1}[:Z(1 sin A)]sin A ass ss i (12) can now be rewritten substituting (19) and (23) to give substituting (24) in (2) It will be seen that this last equation contains functions of 0 and terms in the first order of A, whereas the sine expansion (9) contains only terms in the even powers of sin A. Thus when (i zero, the sine form is evidently better as it contains only even powers of sin A. On the other hand, if 0 zero, then in the tangent expansion the terms in the odd orders of sin A vanish and the coeificients of the even orders of sin A are considerably smaller, thus making the tangent expansion of the design formula best when the center lines of the input and output slide slots are at right angles.

This analysis shows that the form of the Lewis- Nickson multiplier having the input and output slots at right angles should be used where at least one of the inputs has a range symmetrical about zero. The symmetrical input should be put on the rotating slide and the unsymmetrical input on the Vertical slide and the tangent expansion of the design equation used.

On the other hand, if both inputs have unsymmetrical ranges about zero, then the computer should be designed with the fixed input slot at an angle 0 to the output slot and the sine expansion of the design formula should be used with 0 equal to the mean value of 1 representing the input on the rotating slide. Where 0 so chosen is very small, there will be a point at which the larger coeflicient of sine A will overbalance the term sin 0 sin A, and, where one range of input values is very slightly asymetrical, the tangent expansion will be superior. The point where the tangent expansion of the design formula becomes preferable to the sine expansion will be close to the point where where A is half the range of Fig. 3 shows a structure for removing the residual error of the device shown in Fig. 1. A frame member is added at an angle to the slot 24 and formed with a slot 46 carrying a slide 41. A link 48 connects the pin 32 on the slide 31 to a pin 50 on the slide 41. The slide 41 moves a distance that is a function of the residual error of the multiplier computed by the sine expansion of the design formula and is so adjusted as to add a, correction factor to exactly correct for this error in the output. The details of the design of a one-bar linkage of this type may be found in a text such as Antonin Svobodas Computing Mechanisms and Linkages, vol. 27 of the radiation series. Other means, such as a correction cam, or cam slot, could also be used to introduce this correction into the output.

This invention is not limited to the particular details of construction, materials and processes described, as many equivalents will suggest themselves to those skilled in the art. It is accordingly desired that the appended claims be given a broad interpretation commensurate with the scope of the invention within the art.

What is claimed is:

1. A linkage operator comprising a first slidably mounted input member having a straight line path, a second pivotally mounted rotative input member with its pivot on the path of the first member, slide means on said second member causing movement along a straight line through said pivotal mounting, a connecting link engaging said first member and slide means; slidably mounted output means disposed to move along a path at an angle to the path of the first input .slide :member :difieringimm ninety adegmes by twice the desired range of :rotation "of it-H8110- tating tslideacarrying input remember :and .2, nonnecting link engaging said slide means and ;outputzmeans.

'2. A linkage operator comprising a vfirst slidablymounted input member, :a second pivotally mounted rotative input member, slide means on sai'dssecond member causing movement along a line through saidpivotal mounting, a connecting link engaging saidfirst member and slide means; slidably "mounted output means free :to :move along a path at an angle to thepath :of the "first input "member differing from ninety :degrees by twice therange of rotation of the rotative input member, a connecting link tengagingsaid slide means and output means; aslot positionedat .an

angle to the path 01' the first output means, "a second output member free to travel in said slot and a link joining the first and second "output members.

"EUGENE W. PIKE.

References Cited in theme of this patent UNITED STATES PATENTS Number Name Date 2,498,312 Svoboda Feb. 21, 1950 2,543,872 Schaefer, Jr Mar. "6, 1951 FOREIGN PATENTS Number Country Date 408,803 Great Britain Apr. 19, 1934 

